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On the topological product of discrete $$\lambda$$-compact spaces. (English) Zbl 0114.14102
General Topology and its Relations to modern Analysis and Algebra, Proc. Sympos. Prague 1961, 148-151 (1962).
[For the entire collection see Zbl 0111.35001.]
A topological space $${\mathfrak X}$$ is said to be $$\kappa$$-compact if every class $${\mathcal M}$$ of closed subsets of $${\mathfrak X}$$ with void intersection contains a subclass $${\mathcal M}' \subseteq {\mathcal M}$$ having a void intersection and a power $$\overline{\overline{\mathcal M'}}$$ with $$\overline {\overline{\mathcal M'}}< \aleph_\kappa$$.
For each cardinal number $$m$$ and each pair of ordinal numbers $$\lambda,\kappa$$, one uses the abbreviation $$\top(m,\lambda) \to \kappa$$ of the statment ”if $${\mathcal F}$$ is a class of discrete $$\lambda$$-compact topological spaces with the power $$\overline{\overline{\mathcal F}}=m$$ then the topological product of the elements of $${\mathcal F}$$ is $$\kappa$$-compact”. The authors give an outline of the proof of the theorem ”if $$\alpha, \gamma$$ are ordinals such that $$\aleph_{\alpha+\gamma}$$ is singular and $$cf(\gamma) < \omega$$ then the statement $$\top(\aleph_{\alpha+\gamma},\alpha +1) \to \alpha+\gamma$$ is false” (using the generalized continuum-hypothesis); they discuss some related problems, too. By the theorem the question ”$$\top (\aleph_{\omega},1) \to \omega$$?”, stated in another paper of the authors [see Acta Math. Acad. Sci. Hung. 12, 87-123 (1961; Zbl 0201.32801)], is answered negative.
Reviewer: G.Grimeisen

##### MSC:
 54D45 Local compactness, $$\sigma$$-compactness 54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
topology